1.68: Another Look at the Wigner Function

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The Wigner function, W(x,p), is a phase space distribution function which behaves similarly to the coordinate \(\left(|\Psi(x)|^{2}\right)\) and momentum \(\left(|\tilde{\Psi}(p)|^{2}\right)\) distribution functions. For example, its integral over phase space is normalized.

\[
\iint W(x, p) d x d p=1
\nonumber \]

In phase space, position and momentum are represented by multiplicative operators, so the calculation of their expectation values has a classical appearance. This, naturally, is part of the appeal of phase space quantum mechanical calculations.

\[
\langle x\rangle=\iint x W(x, p) d x d p
\nonumber \]

\[
\langle p\rangle=\iint p W(x, p) d x d p
\nonumber \]

While the Wigner function is real, unlike \(|\Psi(x)|^{2}\) and \(|\tilde{\Psi}(p)|^{2}\), it can take on negative values making it impossible to interpret it as a genuine probability distribution function. For this reason it is frequently referred to as a quasi-probability function, and loses some of its classical appeal. In any case, the Wigner function is redundant in the sense that it is generated from a Schrödinger coordinate or momentum wave equation.

In what follows, the quantum mechanical Wigner distribution function will be rationalized by reference to familiar classical concepts, such as position, momentum and trajectory.

In classical physics, a trajectory is a temporal sequence of position and momentum states. Let us try to represent a classical trajectory in a quantum mechanical formalism. Suppose a quantum mechanical object, a quon (thank you Nick Herbert), in state |\(\Psi\)> moves from position x –s/2 to position x + s/2. We might represent this transition quantum mechanically as the product of two coordinate space probability amplitudes (reading from left to right).

\[
\langle x- \frac{s}{2} | \Psi\rangle\langle\Psi | x+ \frac{s}{2} \rangle
\nonumber \]

Thus far we have a coordinate representation of a transition from one spatial location to another. However, a phase space description also requires a dynamic (or motional) parameter such as momentum. We can introduce momentum by first rearranging the above product of amplitudes as follows.

\[
\langle\Psi | x+ \frac{s}{2} \rangle\langle x- \frac{s}{2} | \Psi\rangle
\nonumber \]

This convolution of positional states takes on the coherent character of a trajectory with the insertion of the following momentum projector (see Feynman Lectures Volume 3) coupling the two spatial states.

\[
\langle x+ \frac{s}{2} | p\rangle\langle p | x- \frac{s}{2} \rangle
\nonumber \]

This gives us a quantum trajectory expressed in the following product of Dirac brackets,

\[
\langle\Psi | x+ \frac{s}{2} \rangle\langle x+ \frac{s}{2} | p\rangle\langle p | x- \frac{s}{2} \rangle\langle x- \frac{s}{2} | \Psi\rangle
\nonumber \]

The four Dirac brackets are read now from right to left as follows: (1) is the amplitude that a particle in the state \(\Psi\) has position (x - \(\frac{s}{2})); (2) is the amplitude that a particle with position (x - \(\frac{s}{2}\)) has momentum p; (3) is the amplitude that a particle with momentum p has position (x + \(\frac{s}{2}\)); (4) is the amplitude that a particle with position (x + \(\frac{s}{2}\)) is (still) in the state \(\Psi\).

Integration over s yields the Wigner distribution function, which is a superposition of all possible quantum trajectories of the state \(\Psi\), which interfere constructively and destructively, providing a quasi-probability distribution in phase space.

\[
\int\langle\Psi | x+ \frac{s}{2} \rangle\langle x+ \frac{s}{2} | p\rangle\langle p | x- \frac{s}{2} \rangle\langle x- \frac{s}{2} | \Psi\rangle d s = \frac{1}{h} \int \Psi(x+ \frac{s}{2})^{*} \exp \left(i \frac{p s}{\hbar}\right) \Psi(x- \frac{s}{2}) d s
\nonumber \]

given that

\[
\langle x+ \frac{s}{2} | p\rangle\langle p | x- \frac{s}{2}\rangle=\frac{1}{\sqrt{h}} \exp \left(i \frac{p(x+ \frac{s}{2})}{\hbar}\right) \frac{1}{\sqrt{h}} \exp \left(-i \frac{p(x-\frac{s}{2})}{\hbar}\right)=\frac{1}{h} \exp \left(i \frac{p s}{\hbar}\right)
\nonumber \]

While the Wigner distribution is more than a quantum mechanical curiosity and plays an important role in current research (see references below), it is also true, as mentioned above, that it is redundant because it is generated from either a coordinate or momentum wave function. In Dan Styer’s words it is useful in exploring the quantum/classical transition, but it does not eliminate quantum weirdness – it simply repackages it (see reference 12).

Having said this it should be acknowledged that the Wigner phase-space distribution has been measured for the double slit experiment using tomographic techniques (see references 17-19).

Literature references to the Wigner distribution function:

E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749 – 759 (1932).
M. Hillery, R. F. O’Connell, M. O. Scully, and E. P. Wigner, “Distribution functions in physics: Fundamentals,” Phys. Rep. 106, 121 – 167 (1984).
Y. S. Kim and E. P. Wigner, “Canonical transformations in quantum mechanics,” Am. J. Phys. 58, 439 – 448 (1990).
J. Snygg, “Wave functions rotated in phase space,” Am. J. Phys. 45, 58 – 60 (1977).
J. Snygg, “Uses of operator functions to construct refined correspondence principle via the quantum mechanics of Wigner and Moyal,” Am. J. Phys. 48, 964 – 970 (1980).
N. Mukunda, “Wigner distribution for angle coordinates in quantum mechanics,” Am. J. Phys. 47, 192 – 187 (1979).
S. Stenholm, “The Wigner function: I. The physical interpretation,” Eur. J. Phys. 1, 244 – 248 (1980).
G. Mourgues, J. C. Andrieux, and M. R. Feix, “Solutions of the Schrödinger equation for a system excited by a time Dirac pulse of pulse of potential. An example of the connection with the classical limit through a particular smoothing of the Wigner function,” Eur. J. Phys. 5, 112 – 118 (1984).
M. Casas, H. Krivine, and J. Martorell, “On the Wigner transforms of some simple systems and their semiclassical interpretations,” Eur. J.Phys. 12, 105 – 111 (1991).
R. A. Campos, “Correlation coefficient for incompatible observables of the quantum mechanical harmonic oscillator,” Am. J. Phys. 66, 712 – 718 (1998).
M. Belloni, M. A. Doncheski, and R. W. Robinett, “Wigner quasi-probability distribution for the infinite square well: Energy eigenstates and time-dependent wave packets,” Am. J. Phys. 72, 1183 – 1192 (2004).
D. F. Styer, et al., “Nine formulations of quantum mechanics,” Am. J. Phys. 70, 288 – 297 (2002).
H-W Lee, “Spreading of a free wave packet,” Am. J. Phys. 50, 438 – 440 (1982).
D. Home and S. Sengupta, “Classical limit of quantum mechanics,” Am. J. Phys. 51, 265 – 267 (1983).
W. H. Zurek, “Decoherence and the transition from quantum to classical,” Phys. Today 44, 36 – 44 (October 1991).
M. C. Teich and B. E. A. Saleh, “Squeezed and antibunched light,” Phys. Today 43, 26 – 34 (June 1990).
Ch. Kurtsiefer, T. Pfau, and J.Mlynek, “Measurement of the Wigner function of an ensemble of helium atoms,” Nature 386, 150-153 (1997).
M. Freyberger and W. P. Schleich, “True vision of a quantum state,” Nature 386, 121-122 (1997).
D. Leibfried, T. Pfau, and C. Monroe, “Shadows and mirrors: Reconstructing quantum states of motion,” Phys. Today 51, 22 – 28 (April 1998).
W. P. Schleich and G. Süssmann, “A jump shot at the Wigner distribution,” Phys. Today 44, 146 – 147 (October 1991).
R. A. Campos, “Correlation coefficient for incompatible observables of the quantum harmonic oscillator,” Am. J. Phys. 66, 712 – 718 (1998).
R. A. Campos, “Wigner quasiprobability distribution for quantum superpositions of coherent states, a Comment on ‘Correlation coefficient for incompatible observables of the quantum harmonic oscillator,’” Am. J. Phys. 67, 641 – 642 (1999).
C. C. Gerry and P. L. Knight, “Quantum superpositions and Schrödinger cat states in quantum optics,” Am. J. Phys. 65, 964 – 974 (1997).
K. Ekert and P. L. Knight, “Correlations and squeezing of two-mode oscillations,” Am. J. Phys. 57, 692 – 697 (1989).
W. B. Case, “Wigner functions and Weyl transforms for pedestrians,” Am. J. Phys. 76, 937 – 946 (2008).
M. G. Raymer, “Measuring the quantum mechanical wave function,” Contemp. Phys. 38, 343 – 355 (1997).
F. Rioux, “Illuminating the Wigner function with Dirac notation,”
F. Rioux, “The Wigner distribution for the double-slit experiment,” www.users.csbsju.edu/~frioux/wigner/DBL-SLIT-NEW.pdf
F. Rioux, “Basic quantum mechanics in coordinate space, momentum space and phase space,”
F. Rioux, “The Wigner distribution for the harmonic oscillator,”
F. Rioux, “The Wigner distribution for the particle in a box,”
F. Rioux, “The time-dependent Wigner distribution for harmonic oscillator transitions,”
F. Rioux, “The Wigner distribution distinguishes between a superposition and a mixture,”